The 'factorial function' is defined as...
(1)
... and Your request [if I undestood correctly...] is to find the Mc Laurin series of . Very well!... let's start from the well known 'inifinite product'...
(2)
... where...
(3)
... is the so called 'Euler's constant'. From (1) and (2) we derive...
(4)
... and from (4)...
(5)
Deriving both terms of (5) we obtain...
(6)
... so that is...
(7)
Deriving both terms of (6) we obtain...
(8)
... so that is...
(9)
... where is the so called 'Riemann zeta function'. Proceeding along the way we find for ...
(10)
... so that the requested McLaurin expansion is...
(11)
The (11) converges for ...
Kind regards